On the Derivative Expansion for the Electromagnetic Casimir Free Energy at High Temperatures

TL;DR

This paper investigates the high-temperature behavior of electromagnetic Casimir free energy, focusing on the role of zero modes and the validity of the derivative expansion in materials with frequency-dependent permittivity and permeability.

Contribution

It analyzes the conditions under which the derivative expansion remains well-defined for the Casimir free energy at high temperatures, considering frequency-dependent material properties.

Findings

Zero modes dominate at high temperatures due to dimensional reduction.

The derivative expansion is well-defined if $\Omega(\omega)$ vanishes at zero frequency.

Non-analyticities can arise depending on material properties at zero frequency.

Abstract

We study the contribution of the thermal zero modes to the Casimir free energy, in the case of a fluctuating electromagnetic (EM) field in the presence of real materials described by frequency-dependent, local and isotropic permittivity ($\epsilon$) and permeability ($\mu$) functions. Those zero modes, present at any finite temperature, become dominant at high temperatures, since the theory is dimensionally reduced. Our work, within the context of the Derivative Expansion (DE) approach, focusses on the emergence of non analyticities in that dimensionally reduced theory. We conclude that the DE is well defined whenever the function $\Omega(\omega)$, defined by $[\Omega(\omega)]^2 \equiv \omega^2\epsilon(\omega)$, vanishes in the zero-frequency limit, for at least one of the two material media involved.